LECTURE 24
Review
• Maximum likeliho o d estimation
– Have model with unknown parameters:
X ∼ pX (x; θ)
– Pick θ that “makes data most likely”
• Reference: Section 9.3
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http://web.mit.edu/subjectevaluation
476
Classical Statistical Inference
max pX (x; θ)
Chap. 9
θ
– Compare to Bayesian MAP estimation:
Outline
in the context of various probabilistic
frameworks, which provide perspective and
a mechanism for quantitative analysis.
• consider
Reviewthe case of only two variables, and then generalize. We
We first
wish to model–theMaximum
relation between
two variables
of interest, x and y (e.g., years
likelihood
estimation
of education and income), based on a collection of data pairs (xi , yi ), i = 1, . . . , n.
For example, –
xi could
be the years
of education and yi the annual income of the
Confidence
intervals
ith person in the sample. Often a two-dimensional plot of these samples indicates
a systematic,• approximately
linear relation between xi and yi . Then, it is natural
Linear regression
to attempt to build a linear model of the form
max pΘ|X (θ | x) or max
θ
θ
pX |Θ(x|θ)pΘ(θ)
pY (y)
• Sample mean estimate of θ = E[X]
Θ̂n = (X1 + · · · + Xn)/n
• 1 − α confidence interval
y ≈ θ0 testing
+ θ1 x,
• Binary hypothesis
+
P(Θ̂−
n ≤ θ ≤ Θ̂n ) ≥ 1 − α,
θ1 are
unknown
parameters to be estimated.
where θ0 and –
Types
of error
In particular, given some estimates θ̂0 and θ̂ 1 of the resulting parameters,
the value yi corresponding
to xiratio
, as predicted
by the model, is
– Likelihood
test (LRT)
∀ θ
• confidence interval for sample mean
ŷi = θ̂0 + θ̂1 xi .
– let z be s.t. Φ(z) = 1 − α/2
�
Generally, ŷi will be different from the given value yi , and the corresponding
difference
ỹi = yi − ŷi ,
�
zσ
zσ
P Θ̂n − √ ≤ θ ≤ Θ̂n + √
≈1−α
n
n
is called the ith residual. A choice of estimates that results in small residuals
is considered to provide a good fit to the data. With this motivation, the linear
regression approach chooses the parameter estimates θ̂ 0 and θ̂1 that minimize
the sum of the squared residuals,
n
n
�
�
(yi − ŷi )2 =
(yi − θ0 − θ1 xi )2 ,
i=1
i=1
over all θ1 and θ2 ; see Fig. 9.5 for an illustration.
Regression
y
Residual
x yi − θˆ0 − θˆ1 xi
(xi , yi )
×
Linear regression
×
• Model y ≈ θ0 + θ1x
×
min
x y = θ̂0 + θ̂1 x
• Solution (set derivatives to zero):
×
×
x + · · · + xn
x= 1
,
n
yx
• Data: (x1, y1), (x2, y2), . . . , (xn, yn)
θ̂1 =
Figure 9.5: Illustration of a set of data pairs (xi , yi ), and a linear model y =
by minimizing
θ̂ 0 + θ̂ 1 x,
+ θθ01, θx1 the sum of the squares of the residuals
•obtained
Model:
y ≈ θ0over
yi − θ 0 − θ1 x i .
min
θ0 ,θ1
n
�
( yi − θ0 − θ 1 x i ) 2
i=1
n
�
1
−
(yi − θ0 − θ1xi)2
2σ 2 i=1
�n
i=1(xi − x)(yi − y)
�n
2
i=1(xi − x)
• Interpretation of the form of the solution
– Assume a model Y = θ0 + θ1X + W
W independent of X, with zero mean
– Likelihood function fX,Y |θ (x, y; θ) is:
�
y + · · · + yn
y= 1
n
θ̂0 = y − θˆ1x
(∗)
• One interpretation:
Yi = θ0 + θ1xi + Wi, Wi ∼ N (0, σ 2), i.i.d.
c · exp
( y i − θ 0 − θ 1 xi )2
θ0 ,θ1 i=1
×
=0
n
�
– Check that
�
�
cov(X, Y ) E (X − E[X])(Y − E[Y ])
�
�
θ1 =
=
var(X)
E (X − E[X])2
– Take logs, same as (*)
– Solution formula for θˆ1 uses natural
estimates of the variance and covariance
– Least sq. ↔ pretend Wi i.i.d. normal
1
�
The world of regression (ctd.)
The world of linear regression
• In practice, one also reports
• Multiple linear regression:
– data: (xi, x�i, xi��, yi), i = 1, . . . , n
– Confidence intervals for the θi
– model: y ≈ θ0 + θx + θ �x� + θ ��x��
– “Standard error” (estimate of σ)
– R2, a measure of “explanatory power”
– formulation:
min
n
�
(yi − θ0 − θxi − θ �x�i − θ ��x��i )2
θ,θ� ,θ �� i=1
• Some common concerns
– Heteroskedasticity
• Choosing the right variables
– Multicollinearity
– model y ≈ θ0 + θ1h(x)
e.g., y ≈ θ0 + θ1x2
– Sometimes misused to conclude causal
relations
– work with data points (yi, h(x))
– etc.
– formulation:
min
θ
n
�
(yi − θ0 − θ1h1(xi))2
i=1
Binary hypothesis testing
Likelihood ratio test (LRT)
• Binary θ; new terminology:
– null hypothesis H0:
X ∼ pX (x; H0)
• Bayesian case (MAP rule): choose H1 if:
P(H1 | X = x) > P(H0 | X = x)
or
[or fX (x; H0)]
P(X = x | H1)P(H1) P(X = x | H0)P(H0)
>
P(X = x)
P(X = x)
or
P(X = x | H1) P(H0)
>
P(X = x | H0) P(H1)
– alternative hypothesis H1:
[or fX (x; H1)]
X ∼ pX (x; H1)
• Partition the space of possible data vectors
Rejection region R:
reject H0 iff data ∈ R
(likelihood ratio test)
• Nonbayesian version: choose H1 if
• Types of errors:
P(X = x; H1)
> ξ (discrete case)
P(X = x; H0)
– Type I (false rejection, false alarm):
H0 true, but rejected
fX (x; H1)
>ξ
fX (x; H0)
α(R) = P(X ∈ R ; H0)
– Type II (false acceptance,
missed detection):
H0 false, but accepted
(continuous case)
• threshold ξ trades off the two types of error
– choose ξ so that P(reject H0; H0) = α
(e.g., α = 0.05)
β(R) = P(X �∈ R ; H1)
2
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